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# Range of a rational polynomial (Tomato subjective – 76)

problem: Find the set of all values of $${m}$$ such that $${\displaystyle {y} = {\frac{x^2-x}{1-mx}}}$$ can take all real values.

solution: $${\displaystyle {y} = {\frac{x^2-x}{1-mx}}}$$
$${\Leftrightarrow}$$ $${\displaystyle{y – myx = x^2 – x}}$$
$${\Leftrightarrow}$$ $${x^2 + x(my – 1) – y = 0}$$
$${\Leftrightarrow}$$ $${\displaystyle{x} = {\frac{1 – my {\pm} {\sqrt{(my-1)^2+4y}}}{2}}}$$
Now $${y}$$ takes all real values if discriminant $${(my-1)^2 + 4y}$$ is allways $${> 0}$$.
So now we have to find the all values of $${m}$$ such that $${(my-1)^2 + 4y}$$ $${> 0}$$ for all $${y}$$ $${\in}$$ |R.